Drought and high salt stress have a great negative impact on potato growth and development. However, the molecular mechanisms by which different tissues and organs of potato plants respond to drought and high salt stress at different stress times lack definition. In this study, we mapped the whole genome of THSs in potato in response to different stresses using RNA-seq and ATAC-seq technologies and constructed the unique and shared transcriptional regulatory networks (TRNs) under different stresses, stress time points, and tissues in potato. The results showed opposite trends for changes in chromatin accessibility and expression of genes under drought and salt stresses. Forty-eight hours and root-specific TRNs were more complex than those of 3 h and leaf, and there were genes with inconsistent drought- and salt-stress-regulated expression only in root-shared TRNs, demonstrating the variability of potato's response to stresses under different tissues and treatment times. These results provide a basis for elucidating the transcriptional mechanisms underlying the specific response of potato to drought or salt stress and the common response to salt and drought stress.